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To convert the logarithmic equation \( \ln x = 5 \) into an equivalent exponential equation, we use the property of natural logarithms. The equation \( \ln x = 5 \) means that \( x \) is the number such that the natural logarithm (base \( e \)) of \( x \) is 5. 

The conversion rule is:
\[
\ln x = y \Rightarrow x = e^y
\]

So, applying this to \( \ln x = 5 \):
\[
x = e^5
\]

### Answer:
The equivalent exponential equation is:
\[
x = e^5
\]

Would you like a more detailed explanation or have any questions?

Here are five related questions to expand your understanding:
1. How do we convert a general logarithmic equation with different bases to an exponential form?
2. What is the approximate value of \( e^5 \)?
3. How do we solve for \( x \) in equations of the form \( e^{x} = k \)?
4. What are some applications of natural logarithms in calculus?
5. How can we use logarithmic properties to simplify expressions like \( \ln(a \cdot b) \)?

**Tip:** Remember that \( \ln x = 5 \) can be rewritten in terms of exponentiation with base \( e \), the base of natural logarithms, making conversions straightforward.

Change the logarithmic equation \( \ln x = 5 \) to an equivalent equation involving an exponent.

Learn how to convert the logarithmic equation \( \ln x = 5 \) into its equivalent exponential form. This solution explains the natural logarithm properties and provides the final answer, suitable for high school students studying logarithmic and exponential functions.

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